Berno Uli Revised

5/21/2018 Berno Uli Revised

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CONSERVATION OF MASS

Mass is conserved even during chemical reactions.

Conservation of mass: Mass, like energy, is a conserved property,

and it cannot be created or destroyed during a process.

Closed sys tems: The mass of the system remain constant duringa process.

Contro l volumes: Mass can cross the boundaries, and so we must

keep track of the amount of mass entering and leaving the control

volume.

Mass m and energy E can be converted to each other:

c is the speed of light in a vacuum, c = 2.9979108m/s

The mass change due to energy change is negligible.


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Conservation of Mass

The conservation of mass relation for a closed systemundergoing a

change is expressed as msys= constantor dmsys/dt =0, which is the

statement that the mass of the system remains constant during aprocess.

Mass balance for a control volume (CV) in rate form:

the total rates of mass flow into

and out of the control volume

the rate of change of mass within thecontrol volume boundaries.

Continuity equation:In fluid mechanics, the conservation of

mass relation written for a differential control volume is usually

called the continuity equation.


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Mass and Volume Flow Rates

The normal velocity Vnfor a

surface is the component of

velocity perpendicular to the

surface.

Point func t ionshave exact differentials

Path fun ct ionshave inexact differentials

The differential mass flow rate

Mass flow rate:The amount of mass flowing

through a cross section per unit time.


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Conservation of Mass Principle

Conservation of mass principle

for an ordinary bathtub.

The conservation of mass principle for a control volume: The net mass transfer

to or from a control volume during a time interval tis equal to the net change

(increase or decrease) in the total mass within the control volume during t.

the total rates of massflow into and out of the

control volume

the rate of change of mass

within the control volume

boundaries.

Mass balance is applicable to

any control volume undergoing

any kind of process.


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Mass Balance for Steady-Flow Processes

Conservation of mass principle for a two-

inletone-outlet steady-flow system.

During a steady-flow process, the total amount of mass contained within a

control volume does not change with time (mCV= constant).

Then the conservation of mass principle requires thatthe total amount of massentering a control volume equal the total amount of mass leaving it.

For steady-flow processes, we are

interested in the amount of mass flowing per

unit time, that is, the mass flow rate.

Multiple inlets

and exits

Single

stream

Many engineering devices such as nozzles,diffusers, turbines, compressors, and

pumps involve a single stream (only one

inlet and one outlet).


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Special Case: Incompressible Flow

During a steady-flow process, volume

flow rates are not necessarily conserved

although mass flow rates are.

The conservation of mass relations can be simplified even further when

the fluid is incompressible, which is usually the case for liquids.

Steady,

incompressible

Steady,

incompressible

flow (single stream)

There is no such thing as a conservation of

volume principle.

However, for steady flow of liquids, the volume flow

rates, as well as the mass flow rates, remain

constant since liquids are essentially incompressible

substances.


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Section 5.3 - Equation For Steady Motion of a Real Fluid

along a stream line

Real fluid with friction Ideal fluid with no friction

Real fluid has the additional term of fluid friction, hf. Fluid friction, hf

is the shear stress at the boundary of the element and is the area overwhich the shear stress acts. Shear stress is


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Formula for hfis based on

PL (inside surface area of the conduit wall), A of the pipe, specific weight

and sheer stress at the wall, o


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Sample Problem 5.3, pg 137

Water flows through a 150-ft long, 9-in-diameter pipe at 3.8 cfs. At the

entry point, the pressure is 30 psi; at the exit point, 15 ft higher than

the entry point, the pressure is 20 psi. Between these two points, find

(a) the pipe friction head loss,

(b) the wall shear stress, and

(c) the friction force on the pipe.


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Section 5.4Pressure in Conduits of Uniform Cross SectionStagnation pressureis the pressure a fluid exerts when it is forced to stop

moving. Consequently, although a fluid moving at higher speed will have a

lower static pressure, it may have a higher stagnation pressure when forcedto a standstill.

The centerline in Figure 4.12 shows that the velocity becomes zero at the

stagnation point.
http://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Stagnation_pressure


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Static, Dynamic, and Stagnation PressuresThe kinetic and potential energies of the fluid can be converted to flow

energy (and vice versa) during flow, causing the pressure to change.

Multiplying the Bernoulli equation by the density gives

Total pressure:The sum of the static, dynamic, and

hydrostatic pressures. Therefore, the Bernoulli equation

states that the total pressure along a streamline is constant.

P is the static pressure:It does not incorporate any dynamic effects; it

represents the actual thermodynamic pressure of the fluid. This is the same

as the pressure used in thermodynamics and property tables.

V2/2 is the dynamic pressure:It represents the pressure rise when the

fluid in motion is brought to a stop isentropically.

gz is the hydrostatic pressure:It is not pressure in a real sense since its

value depends on the reference level selected; it accounts for the elevation

effects, i.e., fluid weight on pressure. (Be careful of the signunlikehydrostatic pressure gh which increases with fluid depth h, the hydrostatic

pressure term gz decreases with fluid depth.)


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Stagnation pressure: The sum of the static and dynamic pressures. It represents

the pressure at a point where the fluid is brought to a complete stop isentropically.

Close-up of a Pitot-static probe,

showing the stagnation pressure hole

and two of the five static circumferential

pressure holes.

The static, dynamic, and

stagnation pressures measured

using piezometer tubes.


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Streaklines produced by

colored fluid introduced

upstream of an airfoil; sincethe flow is steady, the

streaklines are the same as

streamlines and pathlines.

The stagnation streamline

is marked.

Careless drilling of

the static pressure

tap may result in an

erroneous reading

of the static

pressure head.


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Page 140, Example Problem 5.4.1Find the stagnation pressure on the

nose of a submarine moving at 12 knots is seawater ( = 64 lb/ft3) when it

is 70 ft below the surface.


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THEBERNOULLI EQUATION

Bernoulli equation:An approximate relation between pressure,

velocity, and elevation, and is valid in regions of steady,

incompressible flow where net frictional forces are negligible.

Despite its simplicity, it has proven to be a very powerful tool in fluid

mechanics.

The Bernoulli approximation is typically useful in flow regions outside

of boundary layers and wakes, where the fluid motion is governed bythe combined effects of pressure and gravity forces.

The Bernoulli equation is an

approximate equation that is valid

only in inviscid regions of flowwhere net viscous forces are

negligibly small compared to

inertial, gravitational, or pressure

forces. Such regions occur

outside of boundary layers and

wakes.


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Limitations on the Use of the Bernoulli Equation

1. Steady flowThe Bernoulli equation is

applicable to steady flow.

2. Frictionless flowEvery flow involves

some friction, no matter how small,

and frictional effects may or may not

be negligible.

3. No shaft workThe Bernoulli equation

is not applicable in a flow section that

involves a pump, turbine, fan, or any

other machine or impeller since such

devices destroy the streamlines and

carry out energy interactions with the

fluid particles. When these devices

exist, the energy equation should be

used instead.

Frictional effects, heat transfer, andcomponents that disturb the

streamlined structure of flow make the

Bernoulli equation invalid. It should

not be used in any of the flows shown

here.


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Limitations on the Use of the Bernoulli Equation

4. Incompressible flowDensity is taken

constant in the derivation of the

Bernoulli equation. The flow isincompressible for liquids and also by

gases at Mach numbers less than

about 0.3.

5. No heat transferThe density of a gas

is inversely proportional to temperature,

and thus the Bernoulli equation should

not be used for flow sections that

involve significant temperature change

such as heating or cooling sections.

6. Flow along a streamlineStrictly

speaking, the Bernoulli equationisapplicable along a streamline. However,

when a region of the flow is irrotational

and there is negligibly small vorticity in

the flow field, the Bernoulli equation

becomes applicable across streamlines

as well.

Frictional effects, heat transfer, and

components that disturb the

streamlined structure of flow make

the Bernoulli equation invalid. It

should not be used in any of the

flows shown here.


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When the flow is irrotational, the Bernoulli equation becomes applicable

between any two points along the flow (not just on the same streamline).

i r rotat ionalmeans not having rotation or the vorticity has the

magnitude zero everywhere.


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Kinetic EnergyVelocity head

Potential EnergyElevation head

Flow Energy - Pressure head = P/ or P/g

Total head = Velocity head + Elevation head + Pressure head

The Bernoulli equation states that the sum of the kinetic, potential, and flow

energies of a fluid particle is constant along a streamline during steady flow.

The sum of the kinetic, potential, and

flow energies of a fluid particle isconstant along a streamline during

steady flow when compressibility and

frictional effects are negligible.


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Hydraulic Grade Line (HGL) and Energy Grade Line (EGL)

It is often convenient to represent the level of mechanical energy graphically using

heightsto facilitate visualization of the various terms of the Bernoulli equation.

Dividing each term of the Bernoulli equation by g gives

An alternative form of the

Bernoulli equation is expressed

in terms of heads as: The sum

of th e pressure, veloci ty , and

elevat ion heads is con stant

along a streamlin e.

P/g is the pressure head;it represents the height of a fluid column

that produces the static pressure P.

V2/2g is the velocity head;it represents the elevation needed for afluid to reach the velocity V during frictionless free fall.

z is the elevation head;it represents the potential energy of the fluid.


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The hydraulic

grade line (HGL)

and the energy

grade line (EGL)

for free dischargefrom a reservoir

through a

horizontal pipe

with a diffuser.

Hydraulic grade line (HGL), P/g + z The line that represents the sum of

the static pressure and the elevation heads.

Energy grade line (EGL), P/g + V2/2g + z The line that represents thetotal head of the fluid.

Dynamic head,V2/2g The difference between the heights of EGL and HGL.


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For stationary bodiessuch as reservoirs or lakes, the EGL and HGL coincide with

the free surface of the liquid.

The EGL is always a distance V2/2g above the HGL. These two curves approach

each other as the velocity decreases, and they diverge as the velocity increases.

In an idealized Bernoulli-type flow, EGL is horizontal and its height remains

constant.

For open-channel flow, the HGL coincides with the free surface of the liquid, and

the EGL is a distance V2/2g above the free surface.

At apipe exit, the pressure head is zero (atmospheric pressure) and thus theHGL coincides with the pipe outlet.

The mechanical energy lossdue to frictional effects (conversion to thermal

energy) causes the EGL and HGL to slope downward in the direction of flow. The

slope is a measure of the head loss in the pipe. A component, such as a valve,

that generates significant frictional effects causes a sudden drop in both EGL and

HGL at that location.

A steep jump/dropoccurs in EGL and HGL whenever mechanical energy is

added or removed to or from the fluid (pump, turbine).

The (gage) pressure of a fluid is zero at locations where the HGL intersects the

fluid. The pressure in a flow section that lies above the HGL is negative, and the

pressure in a section that lies below the HGL is positive.

Notes on HGL and EGL


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In an idealized Bernoulli-type flow,EGL is horizontal and its height

remains constant. But this is not

the case for HGL when the flow

velocity varies along the flow.

A steep jump occurs in EGL and HGL

whenever mechanical energy is added to

the fluid by a pump, and a steep drop

occurs whenever mechanical energy is

removed from the fluid by a turbine.

The gage pressure of a fluid is zero at

locations where the HGL intersects the

fluid, and the pressure is negative

(vacuum) in a flow section that lies

above the HGL.


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Example:

Spraying Water

into the Air

Example: Water Discharge

from a Large Tank


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Example: Siphoning OutGasoline from a Fuel Tank


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Example: Velocity Measurement

by a Pitot Tube


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Example: The Rise of the

Ocean Due to a Hurricane

The eye of hurricane Linda (1997 in

the Pacific Ocean near Baja

California) is clearly visible in thissatellite photo.

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Berno Uli Revised